Inspired by Steve Gardham's posts of 19 Feb 2009 and 18 Aug 2019, I looked up Harry Clifton. He was known as a moralistic singer touring respectable halls in England with a troupe which included "serio-comic" singer Fanny Edwards. A good short biography is at https://folksongandmusichall.com/index.php/harry-clifton/ . That web page includes a published notice from 1866. It lists Clifton's songs, one of which is "Send Back My Barney to Me," specified as a song for Miss Edwards to sing.
Taken together, this all says to me that we've established the composer and the English provenance of the original song. From there, I'd say it underwent a fairly modest folk process. And aside from Clifton apparently not having any "bring back, bring back" litany, his lyrics would fit today's music quite well.
-Bari Ramsey